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Quaternary Quadratic Residue Codes and Unimodular Lattices

Publication ,  Journal Article
Bonnecaze, A; Solé, P; Calderbank, AR
Published in: IEEE Transactions on Information Theory
January 1, 1995

We construct new self-dual and isodual codes over the integers modulo 4. The binary images of these codes under the Gray map are nonlinear, but formally self-dual. The construction involves Hensel lifting of binary cyclic codes. Quaternary quadratic residue codes are obtained by Hensel lifting of the classical binary quadratic residue codes. Repeated Hensel lifting produces a universal code defined over the 2-adic integers. We investigate the connections between this universal code and the codes defined over Z4, the composition of the automorphism group, and the structure of idempotents over Z4. We also derive a square root hound on the minimum Lee weight, and explore the connections with the finite Fourier transform. Certain self-dual codes over Z4 are shown to determine even unimodular lattices, including the extended quadratic residue code of length q + 1, where q ≡ −1(mod8) is a prime power. When q = 23, the quaternary Golay code determines the Leech lattice in this way. This is perhaps the simplest construction for this remarkable lattice that is known. © 1995 IEEE

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Published In

IEEE Transactions on Information Theory

DOI

EISSN

1557-9654

ISSN

0018-9448

Publication Date

January 1, 1995

Volume

41

Issue

2

Start / End Page

366 / 377

Related Subject Headings

  • Networking & Telecommunications
  • 4613 Theory of computation
  • 4006 Communications engineering
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing
 

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Bonnecaze, A., Solé, P., & Calderbank, A. R. (1995). Quaternary Quadratic Residue Codes and Unimodular Lattices. IEEE Transactions on Information Theory, 41(2), 366–377. https://doi.org/10.1109/18.370138
Bonnecaze, A., P. Solé, and A. R. Calderbank. “Quaternary Quadratic Residue Codes and Unimodular Lattices.” IEEE Transactions on Information Theory 41, no. 2 (January 1, 1995): 366–77. https://doi.org/10.1109/18.370138.
Bonnecaze A, Solé P, Calderbank AR. Quaternary Quadratic Residue Codes and Unimodular Lattices. IEEE Transactions on Information Theory. 1995 Jan 1;41(2):366–77.
Bonnecaze, A., et al. “Quaternary Quadratic Residue Codes and Unimodular Lattices.” IEEE Transactions on Information Theory, vol. 41, no. 2, Jan. 1995, pp. 366–77. Scopus, doi:10.1109/18.370138.
Bonnecaze A, Solé P, Calderbank AR. Quaternary Quadratic Residue Codes and Unimodular Lattices. IEEE Transactions on Information Theory. 1995 Jan 1;41(2):366–377.

Published In

IEEE Transactions on Information Theory

DOI

EISSN

1557-9654

ISSN

0018-9448

Publication Date

January 1, 1995

Volume

41

Issue

2

Start / End Page

366 / 377

Related Subject Headings

  • Networking & Telecommunications
  • 4613 Theory of computation
  • 4006 Communications engineering
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing